Gravitational scattering of a quantum particle and the privileged coordinate system

In gravitational scattering the quantum particle probes the Fourier-transforms of a metric. I evaluate the Fourier-transforms of Schwarzschild metrics in standard, harmonic and other coordinate systems in linear and $G^2-$approximations. In general different coordinate systems lead to different scattering. This opens up the possibility to choose the privileged coordinate system which should lead to scattering in agreement with experiment.